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A condition can be both necessary and sufficient. For example, at present, "today is the Fourth of July" is a necessary and sufficient condition for "today is Independence Day in the United States". Similarly, a necessary and sufficient condition for invertibility of a matrix M is that M has a nonzero determinant.
The corresponding logical symbols are "", "", [6] and , [10] and sometimes "iff".These are usually treated as equivalent. However, some texts of mathematical logic (particularly those on first-order logic, rather than propositional logic) make a distinction between these, in which the first, ↔, is used as a symbol in logic formulas, while ⇔ is used in reasoning about those logic formulas ...
A necessary cause is one without which it would be impossible for an effect to occur, while a sufficient cause is one whose presence guarantees the occurrence of an effect. These concepts are largely based on but distinct from ideas of necessity and sufficiency in logic .
That is, necessary truths depend upon the principle of contradiction." [11] The sufficient reason for a necessary truth is that its negation is a contradiction. [4] Leibniz admitted contingent truths, that is, facts in the world that are not necessarily true, but that are nonetheless true.
Causes may sometimes be distinguished into two types: necessary and sufficient. [19] A third type of causation, which requires neither necessity nor sufficiency, but which contributes to the effect, is called a "contributory cause". Necessary causes If x is a necessary cause of y, then the presence of y necessarily implies the prior occurrence ...
In the previous version, the section on sufficient conditions had reversed which is the condition and which is the thing obtained. If you do it this way, it's hard to tell at a glance the main difference between necessary and sufficient: necessary is obtained-->condition, while sufficient is condition-->obtained.
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Necessary and sufficient condition, in logic, something that is a required condition for something else to be the case; Necessary proposition, in logic, a statement about facts that is either unassailably true (tautology) or obviously false (contradiction) Metaphysical necessity, in philosophy, a truth which is true in all possible worlds